Summary
We perturb a steady flow of an incompressible viscous fluid and derive a necessary and sufficient condition for the marginal cases for monotonie energy stability and stability against small (infinitesimal) disturbances to coincide. Evaluation of this condition in two examples singles out, in terms of the parameters of the problem, the cases where necessary and sufficient conditions for stability coincide and thus the steady flow first becomes unstable, together with the class of perturbations responsible for the instability. The analysis is done within the range of strict solutions of each underlying problem; the precise regularity and existence classes are given in Sec. 0. The examples we treat are plane parallel shear flow with a non-symmetric profile in an infinite rotating layer and the effect of rotation on convection.
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von Wahl, W. Necessary and sufficient conditions for the stability of flows of incompressible viscous fluids. Arch. Rational Mech. Anal. 126, 103–129 (1994). https://doi.org/10.1007/BF00391555
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DOI: https://doi.org/10.1007/BF00391555